Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations

Xiaopeng Zhao; Yong Zhou

doi:10.3934/dcdsb.2020142

Article Contents

2021, Volume 26, Issue 2: 795-813. Doi: 10.3934/dcdsb.2020142

This issue Previous Article Stabilities and dynamic transitions of the Fitzhugh-Nagumo system Next Article Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows

Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations

Xiaopeng Zhao^1, , and
Yong Zhou^2,

1.
College of of Sciences, Northeastern University, Shenyang 110819, P. R. China
2.
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai 519082, P. R. China

^* Corresponding author: Xiaopeng Zhao
^* Corresponding author: Xiaopeng Zhao

Received: January 2018

Revised: January 2020

Early access: May 2020

Published: February 2021

This paper is supported by the National Nature Science Foundation of China (grant No. 11401258), Nature Science Foundation of Jiangsu Province (grant No. BK20140130) and China Postdoctoral Science Foundation (grant No. 2015M581689).

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Abstract

The global well-posedness and large time behavior of solutions for the Cauchy problem of the three-dimensional generalized Navier-Stokes equations are studied. We first construct a local continuous solution, then by combining some a priori estimates and the continuity argument, the local continuous solution is extended to all $ t>0 $ step by step provided that the initial data is sufficiently small. In addition, by using Strauss's inequality, generalized interpolation type lemma and a bootstrap argument, we establish the $ L^p $ decay estimate for the solution $ u(\cdot,t) $ and all its derivatives for generalized Navier-Stokes equations with $ \max\{1,\frac{3+q}6\}<\alpha\leq\frac12+\min\{\frac3q-\frac3p,\frac3{2p}\} $.

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Mathematics Subject Classification: Primary: 35B65; Secondary: 35Q35.

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